Orthogonal Morphisms

A pair of morphisms e : A → B , m : X → Y : 𝑒 → 𝐴 𝐵 𝑚 : → 𝑋 𝑌 e:{{A}\to{B}},m:{{X}\to{Y}} are said to be orthogonal, denoted e ⟂ m perpendicular-to 𝑒 𝑚 e\perp m if for every commutative square

A𝐴{A}X𝑋{X}B𝐵{B}Y𝑌{Y}e𝑒\scriptstyle{e}m𝑚\scriptstyle{m}

There exists a unique diagonal filler B → X → 𝐵 𝑋 {{B}\to{X}} that makes the square commute.

A𝐴{A}X𝑋{X}B𝐵{B}Y𝑌{Y}e𝑒\scriptstyle{e}m𝑚\scriptstyle{m}∃!\scriptstyle{\exists!}

We will occasionally say that e 𝑒 e is left orthogonal to m 𝑚 m or m 𝑚 m is right orthogonal to e 𝑒 e , though this terminology is typically reserved for when we generalize to orthogonality with respect to a class of maps.