I am to determine some groups and maps in a diagram. I obviously only have to determine the groups up to "structural equivalence" i.e. isomorphism. Is there an equivalent word for structural equivalence in homomorphisms? For example, the $\mathbb{Z} \rightarrow \mathbb{Z}_6$ homomorphisms $x \mapsto x \mod 6$ and $x \mapsto -(x \mod 6)$, mapping $1,2,3,4,5$ to $1,2,3,4,5$ and $5,4,3,2,1$, respectively, are equivalent, in the sense that $1$ is equivalent to $5$, $2$ to $4$ and so on, and these elements are equivalent in the sense that there is an automorphism $x \mapsto -x$ on $\mathbb{Z}_6$, mapping $1,2,3,4,5$ to $5,4,3,2,1$.
In other words, two $A \rightarrow B$ homomorphisms $\phi_1$ and $\phi_2$ can be said to be structurally equivalent, in the same sense that groups are isomorphic, if there exists some automorphism $\psi$ on $B$ s.t. $\phi_2 = \psi \circ \phi_1$. Is there a word for this?
You can just call them equivalent or congruent. Or conjugate, in the case of endomorphisms.