Let $\sf C$ be a category with all small limits and colimits. Let $*$ be the terminal object, and denote by $\sf C_*$ the under category $*/\sf C$.
Define the functor $I:\sf C\to C_*$ by setting $I(c)$ as the canonical morphism $*\to (*\sqcup c) $ into the coproduct. I read in Hovey's Model Categories that, in case $\sf C$ has a model structure, $I$ is an embedding, i.e. it is (1) faithful and (2) injective on objects.
(1) Take two morphisms $f,g:c'\to c$ in $\sf C$. Let $i_c,i_*$ be the canonical morphisms into $*\sqcup c$; then the components of $I(f)$ are $(i_c\circ f, i_*)$, and similarly those of $I(g)$ are $(i_c\circ g, i_*)$. Hence $I_{c',c}$ is injective iff $i_c$ is monic and, in order to prove that $I$ is faithful, I'd prove that the canonical morphism $c\to (*\sqcup c)$ is monic for any object $c$. I don't have a clue on how to use the model structure to get this done.
(2) Let $c,c'$ be objects such that $* \sqcup c=* \sqcup c'$, in a way that the canonical morphisms $*\to * \sqcup c$ and $*\to * \sqcup c'$ are equal too. I shall prove that $c=c'$, which seems quite strong; are you sure that I shouldn't prove just $c\cong c'$? Anyway, I can't even obtain a morphism $c\to c'$, let alone an isomorphism.
I'm new to model categories but I have a background in category theory, and this problem was at the beginning of Hovey's book (like at page 4). So I guess these problems should be easy checks for the reader, while I don't even know how to start them; is this because I should read other texts before Hovey, or should these checks be easy for anyone practical with category theory? Thanks for any help.
I add below the paragraph of the book (the last of page 4) which I quoted, to be sure that it's not a misunderstanding of mine.
The statement isn't true. Any category admits a model structure, so if the statement were true then $\mathcal{C} \to \mathcal{C}_{*}$ would be faithful and injective on objects for any complete and cocomplete category $\mathcal{C}$. But this isn't true. For example, if $\mathcal{C}$ is the category of commutative rings then $\mathcal{C}_{*}$ is the terminal category, and the unique functor $\mathcal{C} \to \mathcal{C}_{*}$ is neither faithful nor injective on objects.