The free category \(3 := \mathbf{Free}(\boxed{\overset{v_1}\bullet \xrightarrow{f_1}\overset{v_2}{\bullet}\xrightarrow{f_2}\overset{v_3}{\bullet}})\) has three objects and six morphisms. Give the morphisms names and write out the composition operation in a 6x6 matrix. Which are the identities?
Identities are 1,2,3
| \(\circ\) | 1 | 2 | 3 | f1 | f2 | f12 |
|---|---|---|---|---|---|---|
| 1 | 1 | f1 | f12 | |||
| 2 | 2 | f2 | ||||
| 3 | 3 | |||||
| f1 | f1 | f12 | ||||
| f2 | f2 | |||||
| f12 | f12 |