I am working on an exercise on what I have homomorphisms $\psi, \eta_1,\eta_2,\phi$ s.t. the diagram commutes:
$$\require{AMScd} \begin{equation}\begin{CD} R_1 @>\psi>> R_2\\ @VV{\eta_1}V @VV{\eta_2}V\\ R_3 @>\phi>> R_4 \end{CD}\end{equation}$$
I'd like to find a homomorphism
$$ \require{AMScd} \begin{equation}\begin{CD}R_3/Im(\eta_1) @>\overline{\phi}>> R_4/Im(\eta_2) \end{CD}\end{equation}$$
When and how can I do this?
Many thanks in advance!
Let $\pi$ be the canonical projection $R_4 \to R_4 / \operatorname{im}(\eta_2)$. Then $\phi$ induces the desired homomorphism if $\operatorname{im}(\eta_1) \subset \ker(\pi \circ \phi)$, or equivalently, $\pi \circ \phi \circ \eta_1 = 0$. The commutativity of the diagram gives the last statement.