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Lemma 1.6.11. Suppose $f_1,...,f_n$ is a composable sequence - a "path" - of morphisms in a category. If the composite $f_kf_{k-1}...f_{i+1}f_i$ equals $g_m...g_1$ for another composable sequence of morphsism $g_1,...,g_m$, then $f_n...f_1 = f_n...f_{k+1}g_m...g_1f_{i-1}...f_1$

... In such cases, Lemma 1.6.11 and transitivity of equality implies that commutativity of the entire diagram may be checked by establishing commutativity of each minimal subdiagram in the directed graph. Here, a minimal subdiagram corresponds to a composition relation $h_n...h_1 = k_m...k_1$ that cannot be factored into a relation between shorter paths of composable morphisms. The graph corresponding to a minimal relation is a "directed polygon" with a commutative triangle being the simplest case.

This is from a book "Category Theory in Context" by Emily Riehl. As an example, the author gives the case of a commutative cube (a cube of objects and morphisms in a category) such as this:enter image description here

The formal definition of a diagram in a category $\mathcal{C}$ is that it is a functor $F\colon\mathcal{I}\to\mathcal{C}$ for some category $\mathcal{I}$, which is called the shape of the diagram. It is defined the same way in the aforementioned book.

What I wish to know if whether we can formalize the aforementioned concept of a "minimal subdiagram" as it is called by Riehl with respect to an aribtrary functor $D\colon\mathcal{I}\to\mathcal{C}$ considered as a diagram of shape $\mathcal{I}$ in a category $\mathcal{C}$.

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I don't understand why the index notation is used. Why not say:

If $f$ is a morphism and $f = g\circ h$, then $a \circ f \circ b = a \circ g \circ h \circ b$, wherever $a \circ f \circ b$ is defined.

Since we're in a category, this is equivalent to the lemma given. To reach any path expandable-to from $f$ you just recursively apply this lemma in-place.

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