punctuation on mathematical formulas
A standard principle for the punctuation of mathematical formulae is that they should be punctuated exactly the same as text is. Even when the formula is displayed in a separate line, it should follow standard punctuation: it should end with a dot, it should not necessarily be preceded by a colon, it should not be outside of the sentence.
Bad punctuation Let $f \colon \mathbb{R} \to \mathbb{R}$ be defined by the following formula: $$f(x) = e^{x}$$ A duoidal category has a distributor $$d \colon (X \lhd W) \otimes (Y \lhd Z) \to (X \otimes Y) \lhd (W \otimes Z)$$
Better punctuation Let $f \colon \mathbb{R} \to \mathbb{R}$ be defined by $$f(x) = e^{x}.$$ A duoidal category has a distributor, $$d \colon (X \lhd W) \otimes (Y \lhd Z) \to (X \otimes Y) \lhd (W \otimes Z).$$
As a general principle, the reader should be able to read the sentence out loud on their head, reading each mathematical symbol out loud and reaching a well-punctuated sentence.
References.