Let $\mathbf C$ a category with an initial object named $0$.
Is there a name for the pair of arrows $f,g\colon A\to B$ such that the unique arrow $0\to A$ is their equalizer? And dually, is there a special name for $f,g\colon A\to B$ such that the coequalizer is $B\to 1$, when $1$ is the terminal object of $\mathbf C$? Finally, is it useful to name them? :)
I can figure out how it works in case $\mathbf C$ is concrete: I want to map the fact that a couple of arrows is "everywhere equal" (if "coker(f,g)=the whole") or "nowhere equal" (if "ker(f,g)=nothing unnecessary").
No ideas for general situation + I'm searching references (something make me think about Lawvere but I'm not able to recover anything).
Thanks a lot!