Connecting Homomorphism is Functorial

Theorem
Let $A$ be a commutative ring with unity.

Let:


 * $\begin{xy}\xymatrix{

&&& M_1 \ar@{->}[rr] \ar@{->}[dl] \ar@{->}[dd]|!{[d];[d]}\hole && M_2 % \ar@{->}[rr] \ar@{->}[dl] \ar@{->}[dd]|!{[d];[d]}\hole && M_3 \ar@{->}[dl] \ar@{->}[dd]|!{[d];[d]}\hole \ar@{->}[rr] && 0 \\ && M_1' \ar@{->}[rr] \ar@{->}[dd] && M_2' \ar@{->}[rr] \ar@{->}[dd] && M_3' \ar@{->}[dd] \ar@{->}[rr] && 0 \\ & 0 \ar@{->}[rr]|!{[r];[r]}\hole && N_1 \ar@{->}[rr]|!{[r];[r]}\hole \ar@{->}[dl]_{F’} && N_2 \ar@{->}[rr]|!{[r];[r]}\hole \ar@{->}[dl] && N_3 \ar@{->}[dl]_{F’’} \\ % 0 \ar@{->}[rr] && N_1' \ar@{->}^(.65){e’}[rr] && N_2' \ar@{->}[rr] && N_3' }\end{xy}$ be a commutative diagram of $A$-modules.

Suppose that the rows are exact.