I have a given map $\Phi: \mathcal{G}\longrightarrow \mathcal{H}$ between two groupoids such that $\Phi_g: \mathcal{G}_x\longrightarrow \mathcal{G}_y$ is a functor between the groupoids $\mathcal{G}_x$ and $\mathcal{G}_y$ for each arrow $g: x\longrightarrow y$.
For this questions, it is not relevant how the above was obtained.
The map $\Phi$ is not a groupoid homomorphism but for each pair of arrows $g: x\longrightarrow y$ and $h: y\longrightarrow z$ there is a natural transformation $$\eta_{h, g}: \Phi_{h\circ g}\Longrightarrow \Phi_h\circ \Phi_g.$$ How can I realize this family of natural transformation as a global map? By global map I mean that I would like to understand $\eta$ as a map:
$$\eta: \mathcal{G}\times \mathcal{G}\longrightarrow \textrm{Something}?$$
Thanks.
Remark. I would like to avoid 2-categories.
The best answer is that $\Phi_{\_}$ is a oplax (2-)functor from $\mathcal{H}$ to the 2-category of groupoids.* $\eta$ is the structural map that's analogous to the equation $F(f \circ g) = F(f) \circ F(g)$ for ordinary functors. It's the reason that the functor is oplax: we only have a morphism (not an isomorphism) $\Phi_{h\circ g}\Longrightarrow \Phi_h\circ \Phi_g$. Were this in the other direction, you'd have a lax functor, and if it were an isomorphism, you'd have a pseudofunctor (all subject to the other various coherences that 2-functors have to satisfy).
There really isn't a good way of avoiding talking about 2-categories here. Much like you can't just ignore natural transformations when talking about ordinary functors, we can't ignore the 2-categorical structure of the category of groupoids. This is made apparent by the fact that $\eta$ isn't just an equality: it's a 2-cell in the 2-category of groupoids (a natural transformation between functors).
However, if you aren't really looking for insight, you could say (mechanically) that $\eta$ is, for each $f, g \in \mathcal{Mor}(\mathcal{H})$, a natural transformation $\Phi_{h\circ g}\Longrightarrow \Phi_h\circ \Phi_g$. That is, $\eta$ is in the set
$$ \prod_{f, g \in \mathcal{Mor}(\mathcal{H})} [\mathcal{G}_x, \mathcal{G}_z](\Phi_{h\circ g}, \Phi_h\circ \Phi_g) $$
(where $x$ and $z$ are the source of $g$ and the target of $h$ respectively). $\eta$ presumably satisfies some additional coherances, so it's in a better behaved subset of that set.
*A bit of abuse of notation here: $\Phi_x$ would be $\mathcal{G}_x$ and $\Phi_g$ would be the maps you referred to.