Simple examples of initial/terminal morphisms that aren’t limits/colimts

Question

The wikipedia page on universal properties defines initial and terminal morphisms, and then gives three examples: Tensor algebras, products, and limits/colimits.

However, products are an example of limits, and I don’t have a good intuition about tensor algebras. Another example that I’ve seen is that of initial/terminal object, but this is perhaps too simple for me to get a good intuition.

Are there basic examples of initial/terminal morphisms that simple and easy to understand, and are not limits/colimits and not just initial/terminal objects?

Answer 1

Every adjunction yields initial and terminal morphisms in the sense described in the page you linked. More precisely, given a functor $F : \mathsf C \to \mathsf D$

$F$ admits a left adjoint if and only if there is an initial morphism from $X$ to $F$ for each $X\in\mathsf D$.

$F$ admits a right adjoint if and only if there is an terminal morphism from $F$ to $X$ for each $X\in\mathsf D$.

So there is a plethora of examples that are not limits/colimits. Nex's example about free monoids in the comments is among them.