# Connecting Homomorphism is Functorial

## Theorem

Let $A$ be a commutative ring with unity.

Let:

$\begin{xy}\xymatrix{ &&& M_1 \[email protected]{->}[rr] \[email protected]{->}[dl] \[email protected]{->}[dd]|!{[d];[d]}\hole && M_2 % \[email protected]{->}[rr] \[email protected]{->}[dl] \[email protected]{->}[dd]|!{[d];[d]}\hole && M_3 \[email protected]{->}[dl] \[email protected]{->}[dd]|!{[d];[d]}\hole \[email protected]{->}[rr] && 0 \\ && M_1' \[email protected]{->}[rr] \[email protected]{->}[dd] && M_2' \[email protected]{->}[rr] \[email protected]{->}[dd] && M_3' \[email protected]{->}[dd] \[email protected]{->}[rr] && 0 \\ & 0 \[email protected]{->}[rr]|!{[r];[r]}\hole && N_1 \[email protected]{->}[rr]|!{[r];[r]}\hole \[email protected]{->}[dl]_{F’} && N_2 \[email protected]{->}[rr]|!{[r];[r]}\hole \[email protected]{->}[dl] && N_3 \[email protected]{->}[dl]_{F’’} \\ % 0 \[email protected]{->}[rr] && N_1' \[email protected]{->}^(.65){e’}[rr] && N_2' \[email protected]{->}[rr] && N_3' }\end{xy}$

be a commutative diagram of $A$-modules.

Suppose that the rows are exact.