equivalence of categories preserves group objects I was wondering if an equivalence of categories preserves group objects?
Intuitively yes but I can't show it.
given a group object $C = (C,\mu,\nu,\sigma)$ (respectively multiplication, unit and inverse) of a category $\cal{C}$, and $\cal{D}$ an equivalent category ($F:{\cal{C}\rightarrow D},G:\cal{D}\rightarrow C$) is $FC$ a group object?
I would like to say that I define $\mu' := F\mu(G\times G)(FC\times FC)$ intuitively this makes sense but $GFC \neq C$. What else could be done? 
 A: Here's the short way to show this. Assume that categories $\mathcal{C}$ and $\mathcal{D}$ have finite products, and let $F : \mathcal C \to \mathcal D$ be any functor that preserves finite products. Since the theory of groups is a Lawvere theory, a group object in a category $\mathcal{E}$ with finite products is the same thing as a finite product preserving functor $\mathcal{T}_{\mathbf{Grp}}\to\mathcal{E}$ where $\mathcal{T}_{\mathbf{Grp}}$ is the Lawvere theory of groups. If $H:\mathcal{T}_{\mathbf{Grp}}\to\mathcal{C}$ is a group object, then so is $F\circ H : \mathcal{T}_{\mathbf{Grp}}\to\mathcal{D}$ since the composition of finite product preserving functors is finite product preserving.
That an equivalence of categories preserves limits (and colimits) and thus finite products in particular, is easiest and most useful to see by rectifying it to an adjoint equivalence at which point it becomes a standard fact about adjoint functors.
If you don't want to assume that all finite products exist in $\mathcal{C}$ and $\mathcal{D}$, then you will still need to assume that in $\mathcal{C}$ all representatives of $C^n$ for each $n$ exists for the notion of group object (on $C$) to make sense. This will imply that the finite product $(FC)^n$ for each $n$ exists in $\mathcal{D}$. I say "representatives" because finite products (and limits/colimits in general) are only determined up to unique isomorphism and so there are many objects that can claim to be, e.g. $C^2$, in particular, both $F(C\times C)$ and $FC\times FC$ can claim to be $(FC)^2$. You can then apply the earlier argument to the full subcategories of $\mathcal{C}$ and $\mathcal{D}$ with finite products generated by $C$ and $FC$ respectively.
Going back to assuming all finite products, there is a natural isomorphism $\varphi: F(A\times B)\to FA\times FB$ natural in $A$ and $B$ and an isomorphism $\psi : F1 \to 1$ witnessing $F$ preserving finite products. Concretely, given a group object $(C,\mu,\nu,\sigma)$, you get a new group object $(FC, F\mu\circ\varphi_{C,C}^{-1},F\nu\circ\psi^{-1},F\sigma)$. Showing that this satisfies the laws of a group object is mostly an exercise in applying the naturality of $\varphi^{-1}$.
