I have a question about the definition of a group object in Category Theory (see the Wikipedia article).
$G$ is an object of $\mathcal{C}$, and we assume that $\mathcal{C}$ admits finite products. The multiplication map $\mu$ is a map from $G \times G$ to $G$. It must obey a certain associativity condition, and this is where my confusion starts.
We identify $(G \times G) \times G$ with $G \times (G \times G)$, and define two functions from $G \times G \times G$ to $G \times G$: one given by $(\text{Id}_G, \mu)$ and one given by $(\mu, \text{Id}_G)$.
What is meant in categorical terms by a map of the form $(\text{Id}_G, \mu)$? The definition of a product means that a map into $X \times Y$ is the same as a map into $X$ together with a map into $Y$. So a map $G\times G \times G \to G \times G$ should consist of two maps $G\times G\times G \to G$, right? But neither $\text{Id}_G$ nor $\mu$ are such maps.
When I think about how this construction works on $\mathsf{Set}$, $\mathsf{Grp}$, and other concrete categories, it seems that a morphism $X \to Z$ can be extended to a morphism $X \times Y \to Z$, by just ignoring the second entry. But what is the categorical notion of this? Or if there isn't one, what are we talking about by $(\text{Id}_G, \mu)$?
The universal property of the product $X \times Y$ is stated in terms of two projection maps $p_X : X \times Y \to X$ and $p_Y : X \times Y \to Y$. Any map $X \to Z$ can be composed with the projection $p_X$ to get a map $X \times Y \to Z$, which does exactly what you'd expect: it only looks at the $X$ variable and it ignores the $Y$ variable.
In addition, the product acts on morphisms as well as objects, so if $f : X \to Y$ and $g : Z \to W$ are two maps, then there is a product map $f \times g : X \times Z \to Y \times W$. The maps you're looking at are built this way, but a bit confusingly, since in the first case $Y = G \times G$ and $X = Z = W = G$ and in the second case $X = G \times G$ and $Y = Z = W = G$.