For a topological space $X$, the fundmental groupoid $\Pi(X)$ has points of $X$ as objects and homotopy classes of paths from $x$ to $y$ as morphisms $x\to y$. A continuous map $f\colon X\to Y$ induces a functor $\Pi(f)\colon \Pi(X)\to\Pi(Y)$ which maps $x \in X$ to $f(x) \in Y$ and a homotopy class $[u]\colon x\to y$ to $[f\circ u]\colon f(x)\to f(y)$.
I don't understand a certain step in a proof of the van Kampen theorem for groupoids in tom Dieck's book Algebraic Topology.
Let $X_0$ and $X_1$ be subspaces of $X$ such that $\mathrm{int}(X_0)\cup \mathrm{int}(X_1) = X$. Let $i_v\colon X_0\cap X_1 \to X_v, j_v\colon X_v\to X$ be inclusions. Then, for any functors $h_v\colon \Pi(X_v)\to\Lambda$ into a groupoid $\Lambda$ such that $h_1\Pi(i_1) = h_0\Pi(i_0)$ there is a unique $\lambda\colon \Pi(X)\to\Lambda$ such that $h_1 = \lambda\Pi(j_1)$ and $h_0 = \lambda\Pi(j_0)$.
In his proof, tom Dieck says that a path $w\colon [a,b]\to U$ represents a morphism $[u]\colon w(a)\to w(b)$ in $\Pi(U)$ where $u = w\circ \alpha$ for some continuous $\alpha\colon [0,1]\to [a,b]$ such that $\alpha(0) = a, \alpha(1) = b$. It can be shown that the homotopy class $[u]$ is independent of the choice of $\alpha$. Moreover, he claims that that if $a = t_0 < t_1 < ... < t_m = b$, then $w$ represents the composition of morphisms $[w{\restriction}_{[t_i,t_{i + 1}]}]$.
Suppose that $w\colon I\to X$ is a path. Then there exists a decomposition $0 = t_0 < t_1 < ... < t_{m + 1} = 1$ such that $w([t_i,t_{i+1}]) \in \mathrm{int}(X_v)$ for some $v = 0,1$. We choose $\gamma\colon \{0,...,m\} \to \{0,1\}$ such that $w([t_i,t_{i + 1}]) \subseteq \mathrm{int}(X_{\gamma(i)})$. Considering $w{\restriction}_{[t_i,t_{i + 1}]}$ as a path $w_i$ in $X_{\gamma(i)}$, we have, by the above, $[w] = \Pi(j_{\gamma(m)})[w_m]\circ ... \circ \Pi(j_{\gamma(0)})[w_0]$. If $\lambda$ exists, then functoriality forces $\lambda[w] = h_{\gamma(m)}[w_m]\circ ... \circ h_{\gamma(0)}[w_0]$.
Defining a functor as follows, it can be shown that $h_{\gamma(m)}[w_m]\circ ... \circ h_{\gamma(0)}[w_0]$ is independent of the choice of the decomposition $0 = t_0 < t_1 < ... < t_{m + 1} = 1$ and $\gamma$. But it still remains to to prove that it is well-defined with respect to homotopy of paths. Here's the relevant excerpt from the book:
From here I'm lost.
What are "edge-paths in the subdivided square"? Can one spell them out concretely? The cited figure doesn't help me.
To what do we "apply $H$" exactly?
How does "differing by a homotopy on some subinterval which stays inside $\mathrm{int}(X_v)$" implies the sameness?
What is the inductive step here exactly?
$(0,0)-(0,2) -(3,2) - (3,3) - (5,3)-(5,5)$
and
$(0,0)-(0,2)-(2,2)-(2,3)-(5,3)-(5,5)$
These two paths are the same, except at one point : at $(2,2)$ the first one continues to $(3,2)$ and then moves up to $(3,3)$, while the second one moves up to $(2,3)$ and then moves right to $(3,3)$ (and then both paths continue on together)
At the point where they differ, there is a homotopy from one to the other that stays entirely within the square $[2,3]\times [2,3]$, so that the homotopy moves almost nothing of the path, and what it moves stays within a little square, hence within $X_0$ or $X_1$ if you apply $H$ to it.
To spell out concretely what the edge paths are, you could say something like : paths that follow the edges of the little squares (which you can define explicitly)
You apply $H$ to these edge paths in $[0,1]\times [0,1]$ and to the little homotopies between said edge paths
Well suppose you have $H$ a homotopy from $\gamma$ to $\gamma'$. You have a sequence of edge paths $\delta_0,...,\delta_n$ such that $H\circ \delta_0 = \gamma$ (up to a constant path at $y$), $H\circ \delta_n = \gamma'$ (same thing) and such that $\delta_i, \delta_{i+1}$ differ only one a small square (as above).
Then let $K$ be a homotopy from $\delta_i$ to $\delta_{i+1}$ which moves only in the square by which they differ. Then find times $0=t_0<....<t_m=1$ that correspond to the various edges of the little squares that $\delta_i,\delta_{i+1}$ are defined on, so that $\delta_{j\mid [t_k,t_{k+1}]} \in X_v$ for $j\in \{i,i+1\}$. Then at all times except one of the times (the one where $\delta_i, \delta_{i+1}$ differ), $\delta_{i\mid [t_k,t_{k+1}]} = \delta_{i+1\mid [t_k,t_{k+1}]}$.
And at the ones they differ, we nonetheless have $[H\circ \delta_{i\mid [t_k,t_{k+1}]}] = [H\circ\delta_{i+1\mid [t_k,t_{k+1}]}]$ in $X_v$ (because the homotopy stays within $X_v$ !!)
So for all $k$, $[H\circ \delta_{i\mid [t_k,t_{k+1}]}] = [H\circ\delta_{i+1\mid [t_k,t_{k+1}]}]$ in $X_v$ (where $v$ is the one that they are inside)
It follows that the thing you defined agrees on $H\circ\delta_{i+1}$ and on $H\circ \delta_i$. So it agrees on $\gamma$ (up to a constant) and $H\circ \delta_1$, then on $H\circ \delta_1$ and $H\circ \delta_2$,.... up to $H\circ \delta_{n-1}$ and $\gamma'$ (up to a constant). So in the end, it agrees on $\gamma$ and $\gamma'$
(to have an intuition, imagine the sequence of edge paths as a snake that starts out as "go horizontally to the end, then vertically to the end" and then wiggles a bit, and each wiggle makes it cross a square, to get to "go vertically to then end and then horizontally to the end"; each wiggle is small enough to ensure that from one wiggle to the other it'll have the same image in $\Lambda$, because when the wiggling part is contained within $X_v$, $h_v$ ensures that the image is the same in $\Lambda$)