Cokernel in universal algebra

Question

Let $(S,f_1,\ldots,f_n)$ be an algebra of some variety and $(T,g_1,\ldots,g_n)$ be another algebra of the same variety. Next let $\varphi:S\to T$ be a homomorphism. I understand well that $\ker\varphi=\{(x,y)\in S^2:\varphi(x)=\varphi(y)\}$, all dandy so far! My question is however for the cokernel, is there a meaningful way to define it in universal algebra. I have tried googling about it and looking in my books but cannot find any definition for it. Does one exist or is it too difficult to define here? I know already in groups it is a bit troublesome.

Answer 1

Yes. Your kernel construction is known in category theory as the kernel pair. There is an exactly categorically dual construction called the cokernel pair. (But depending on what you want to do it may not be the right construction to look at.)