Absolutely not, Since after the coin fabrication it requires some time to stabilize the interface, simply we called aging time. thereafter you start to discharge towards cathodic sweep or 0 V. 

Carbon is an anode，and lithium is the cathode。First lithium should be ionized，so if we discharged lithium ionizes and get inserted into the anode

Thanks @vadivel selvamani 

if first cycle is discharging  what does it means lithium ion goes back to lithium from carbon? Also we do not have first charging to compare with that discharging because charging was happened itself during aging. how can we calculate coloumbic efficiency in first cycle

Thanks, @Vasanth Raj

as you say in discharging lithium ion inserted in to carbon so what is first cycle discharging or charging ?

as vadivel selvamani said first discharge cell to 0V. because it got charged in aging time

Dear Friend,

During the aging time, the electrode surface fully covered/wetted with electrolyte to form an unstable SEI (by observing the OCV after 24h). At the initial discharge process, the lithium ion diffuses to your electrode. Meanwhile, the electrolyte get decomposes to form a stable intact SEI (at ~0.8-0.5 V, depends on electrolyte composition) which is ionically conductive and electrically insulative. the decomposition product contains both inorganic (LiCo3, LiF etc) and organic species. 

Suppose, If I am unable to start the cycling studies what will happen?

the electrolyte continuously get decomposes since the electrode is not covered or covered with labile film leads to electrolyte deficient.  (in EIS, the Rct continuously increasing)

Dear Ashish Gupta,

immediately before your cell fabrication the Li+ ions are recognizable in your electrolyte, only. Immediately after your cell fabrication the Li+ ions extent, also, in both of your (electrolyte//electrodes) interfaces (ONLY). The carbon//electrolyte interface needs some time (6 to 30 minutes) to stabilize, depending, mainly, on the properties and the quantity (%) of the binder. The Li-metal//electrolyte interface as a counter electrode, also, needs some time to stabilize, depending on the quality of your electrolyte and the impurities in your fabrication cell. So, consider an initial time (say about an hour) immediately your cell fabrication as a transition time zone, just wait. Then, you can start discharging your cell.  A proposal method for your case is the GITT[1] Technique. The GITT helps you acknowledge and support better your inquiry learning, questions. If you discharge towards a nominal Voltage value near 0V, then you will acknowledge better the complex phenomena through the Voltage relaxation of the electrodes. If you could form a second Li-metal//electrolyte interface as a reference electrode (as an additional electrode to counter Li electrode), then you will have a 3-electrode cell for a better understanding of the complexity since you will isolate the phenomena in a single (vs. reference) electrode. Upon (discharge/charge) cycling the interfaces become even more complex[2]. The complexity scales with the cycling, not only for the carbon electrode[2,3], but along with the Li-metal electrode[4], also.

1.  Galvanostatic Intermittent Titration Technique :  Electrochemical modeling of intercalation processes with phase field models http://web.mit.edu/ceder/publications/phase_field.pdf  ;   https://partners.metrohm.com/GetDocument?action=get_dms_document&docid=2044489  ;   http://bio-logic.info/assets/comm_tutorials/Batteries_II_ECLab.pdf  ;   Solid State Electrochemical In-Situ Techniques for the Evaluation of Microbatteries  https://link.springer.com/chapter/10.1007/978-1-4899-2263-2_25

2.  Lithium-Ion Batteries and Materials   https://link.springer.com/chapter/10.1007/978-3-662-46657-5_15   ; Potentiostatic and Galvanostatic Intermittent Titration Techniques   http://onlinelibrary.wiley.com/doi/10.1002/0471266965.com125/abstract  ;    https://www.gamry.com/application-notes/battery-research/testing-lithium-ion-batteries/

3.  Electrolytes, SEI and Charge Discharge Kinetics of Li-ion Batteries http://ecst.ecsdl.org/content/25/36/3.full.pdf   ;   Theory of SEI Formation... https://arxiv.org/pdf/1210.3672.pdf

4.  Structural modulation of lithium metal-electrolyte interface with three-dimensional metallic interlayer for high-performance lithium metal batteries   https://www.nature.com/articles/srep30830.pdf

In addition to the excellent answers above, I would like to add that it is easier to understand discharge and change as intercalation / lithiation and deintercalation/delithiation respectively. 

during discharging (half-cell testing for carbon anode) potential increases or decreases in a cycle and WHY?

In half cell, first cycle is discharge ie. lithium intercalate from bulk metallic lithium to carbon anode, while charging the intercalated lithium reverted metallic Li. At carbon anode, during cycling lithium ions are intercalate and deintercalate between the graphene layers of carbon moiety.  Similarly,  Lithium plating and stripping taking place at bulk metallic lithium.