Homomorphism transformations of endomorphisms. I've been trying to figure out if this concept is well defined and if it has a conventional name.
Specifically, suppose we have a homomorphism between two algebraic structures: $$\varphi: R \to S$$
And an endomorphism: $$f: R \to R$$
That satisfies: $$f[\ker (\varphi)] \subseteq \ker(\varphi)$$
It seems, at least to me, that we can define an endomorphism $\bar f$ on $\varphi[R]$ through:
$$(\bar f \circ \varphi) = (\varphi \circ f)$$
And if it would make sense to call $\varphi(f) := \bar f$ so that the above expression can be rewritten in the form:
$$\varphi(f)\varphi(x) = \varphi(fx) \;\;\forall x \in R$$
I am wondering if concepts similar to $\varphi(f)$ are defined in literature, and if it even makes sense to consider it.
 A: Making some assumptions about the algebraic structures in question (so that we can talk about kernels, etc.): since $\ker\varphi$ is invariant under $f$ there is an induced homomorphism $\overline{f}:R/\ker\varphi\to R/\ker\varphi$ making this diagram commute:
$$\require{AMScd}
\begin{CD}
R @>f>> R\\
@V\pi VV @VV\pi V\\
R/\ker\varphi @>>{\overline{f}}> R/\ker\varphi
\end{CD}\tag{1}$$
On the other hand, by the first isomorphism theorem, there is an induced isomorphism $\overline{\varphi}:R/\ker\varphi\to\varphi[R]$ making this diagram commute:
$$\require{AMScd}
\begin{CD}
R @>\varphi>> S\\
@V\pi VV @AA i A\\
R/\ker\varphi @>\cong >{\overline{\varphi}}> \varphi[R]
\end{CD}\tag{2}$$
Your map $\overline{f}$ is just the conjugate of the $\overline{f}$ in (1) by $\overline{\varphi}$ in (2):
$$\require{AMScd}
\begin{CD}
R/\ker\varphi @>{\overline{f}}>> R/\ker\varphi\\
@V\overline{\varphi}V\cong V @V\cong V\overline{\varphi}V\\
\varphi[R] @>>> \varphi[R]
\end{CD}$$
This is how I've typically seen the map arise.
