strict 2-groups VS crossed modules

Let us prove the analogous claim for weak 2-groups (this implies, in particular, the strict case). Let $\mathcal{C}$ be a monoidal category with unit $\mathbb{I} \in \mathcal{C}$.

Claim: Suppose there exists a functor $Inv: \mathcal{C} \to \mathcal{C}$ and natural isomorphisms $\psi_X: \mathbb{I} \stackrel{\cong}{\to} X \otimes Inv(X)$ and $\phi_X: Inv(X) \otimes X \stackrel{\cong}{\to} \mathbb{I}$. Then $\mathcal{C}$ is a groupoid.

Proof: Let $f: X \to Y$ be a map in $\mathcal{C}$. We need to show that $f$ is an isomorphism. Let $g: Y \to X$ be the composed map $$ Y \stackrel{\psi_X \otimes Id_Y}{\to} X \otimes Inv(X) \otimes Y \stackrel{Id_X \otimes Inv(f) \otimes Id_Y}{\to} X \otimes Inv(Y) \otimes Y \stackrel{Id_X \otimes \phi_Y}{\to} X $$ We now claim that $g \circ f$ is an isomorphism. Indeed, $g \circ f$ is equal to the composition $$ X \stackrel{\psi_X \otimes Id_X}{\to} X \otimes Inv(X) \otimes X \stackrel{Id_X \otimes Inv(f) \otimes f}{\to} X \otimes Inv(Y) \otimes Y \stackrel{Id_X \otimes \phi_Y}{\to} X $$ and the naturality of $\phi$ implies that $Inv(f) \otimes f$ is an isomorphism. A similar argument shows that $f \circ g$ is an isomoprhism. It follows that both $f$ and $g$ are isomorphisms.

Pedro, you seem to be making your life difficult! My first suggestion is to read the original sources on this and in particular:

R. Brown and C. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Proc. Kon. Ned. Akad. v. Wet, 79, (1976), 296 – 302, [pdf]

Tracking through from a 2-group to a crossed module and back again should give you a good intuition about what is going on. Something like this: Given a 2-cell $\alpha: g_1\Rightarrow g_2$ in your strict 2-group, you can write $\alpha$ as being $(\alpha \ast_0 g_1^{-1})\ast_0 g_1$ so a pre-whiskering of a 2-cell in the kernel of the source map. This is the sort of 2-cell that comes directly from a crossed module viewpoint, so look at that next.

If $\partial : C\to P$ is a crossed module, then the associated 2-group has the semi-direct product $C\rtimes P$ as its group of 2-cells. In Brown and Spencer you can find the formula for the vertical inverse of a 2-cell $(c,p)$. This gives $(c^{-1},\partial c.p)$, now push this formula back into your original setting multiply out the whiskering and you get a formula for the vertical inverse of $\alpha$.

Of course, this is just what Yonathan has given in non-strict case but relates things back to the original material.

It is worth mentioning once again that the Brown-Spencer method is nicely seen as being a mild but very neat generalisation of the proof that congruences in groups correspond to normal subgroups. It also uses the idea that often (and I do not want to be precise here) an A object in the category of B objects is the same as a B object in the category of A objects.

PS: A good reference for some of this is the short note by Magnus Forrester-Barker: http://arxiv.org/abs/math/0212065.