linguistic:translation, stylistic:tone
Universal property

Just as the limit is a special case of the universal construction, so is the product. Starting with the definition given for the universal property of limits, take J {\displaystyle \mathbf {J} } as the discrete category with two objects, so that C J {\displaystyle \mathbf {C} ^{\mathbf {J} }} is simply the product category C × C. {\displaystyle \mathbf {C} \times \mathbf {C}.} The diagonal functor Δ: C → C × C {\displaystyle \Delta :\mathbf {C} \to \mathbf {C} \times \mathbf {C} } assigns to each object X {\displaystyle X} the ordered pair (X, X) {\displaystyle (X,X)} and to each morphism f {\displaystyle f} the pair (f, f). {\displaystyle (f,f).} The product X 1 × X 2 {\displaystyle X_{1}\times X_{2}} in C {\displaystyle C} is given by a universal morphism from the functor Δ {\displaystyle \Delta } to the object (X 1, X 2) {\displaystyle \left(X_{1},X_{2}\right)} in C × C. {\displaystyle \mathbf {C} \times \mathbf {C}.} This universal morphism consists of an object X {\displaystyle X} of C {\displaystyle C} and a morphism (X, X) → (X 1, X 2) {\displaystyle (X,X)\to \left(X_{1},X_{2}\right)} which contains projections.