Let $R$ be any commutative ring with $1$. I'm having some doubts regarding $R$-Alg, the category of $R$-algebras, which is the co-slice category under $R$ in the category of commutative rings.
(1)Is $R$ initial in the category of $R$-algebras?. I am not convinced by the argument presented in the link regarding the initial object property of $R$. Suppose I take some ring homomorphism $f: R \rightarrow A$. Clearly $f(r)=r.f(1_{R})=r.1_{A}\;.$ Now why can't we have several possible action of $r$ on $1_{A}$, leading to several possible maps. I mean if we consider $R=\mathbb{Z}$ then $f(n)=n.1_{A}$, which is the $n$- times addition of $1_{A}$, therefore the action is uniquely determined and we get the unique $\mathbb{Z}$-algebra map. But it is not clear to me for general $R$.
(2) More generally, is it abelian category? or rather a Grothendieck category like $R$-Mod.
(3) What kind of inclusion is $R\hbox{-}Alg \rightarrow Comrings$? It seems to be a full subcategory. Is there any left or right adjoint to this inclusion?
thanks in advance!
Most of the questions were answered in the comments. Like any forgetful functor from a co-slice category, that from algebras to ring preserves limits but not, generally, colimits-for the latter, just note that tensoring over $R$ is not the same as tensoring over $\mathbb Z$. Thus there can’t be a right adjoint, but there is a left adjoint, given by $A\mapsto R\otimes_{\mathbb Z} A$, with the algebra structure all in the left side of the tensor.