First Isomorphism Theorem/use

Question

How do I apply $T\varepsilon$ to obtain an isomorphism $\sigma_A:\text{coker}T(d_1)\to TA$ by the First Isomorphism Theorem in the snippet below? In fact, I cannot see how FIT applies.

Answer 1

Recall that $\text{coker} (Td_1) = TP_0 \big{/} \text{im}(Td_1)$. By exactness we have $\text{im}(Td_1) = \text{ker}(T_\epsilon)$, so $\text{coker} (Td_1) =TP_0 \big{/} \text{ker}(T_\epsilon) \overset{\sim}{\longrightarrow}\text{im}(T_\epsilon) = T_A$, since $T_\epsilon$ is surjective by exactness of the sequence at $T_A$.