Basepoint in a category

Question

What do they mean in the second paragraph by "takes $v$ to $w$" ?

Answer 1

A morphism $f : (X,v) \to (Y,w)$ that takes $v$ to $w$ is simply a morphism $f : X \to Y$ such that $f \circ v = w$.

This is motivated by the category of sets. A pointed set is a pair $(X,x_0)$ with a set $X$ and a basepoint $x_0 \in X$. We may identify the point $x_0$ with the function $v^X_{x_0} : * \to X, v^X_{x_0}(*) = x_0$. A basepointed preserving function $f : (X,x_0) \to (Y,y_0)$ (i.e. a morphism in the category of pointed sets) is a function $f : X \to Y$ such that $f(x_0) = y_0$. This is the same as $f \circ v^X_{x_0} = v^Y_{y_0}$.