Coproduct of Lawvere Theories

The coproduct 𝒯 1 + 𝒯 2 subscript 𝒯 1 subscript 𝒯 2 \mathcal{T}_{1}+\mathcal{T}_{2} of two Lawvere Theories is the Lawvere theory formed by taking a Pushout

𝔽𝔽{{\mathbb{F}}}𝒯1subscript𝒯1{{\mathcal{T}_{1}}}𝒯2subscript𝒯2{{\mathcal{T}_{2}}}𝒯1+𝒯2subscript𝒯1subscript𝒯2{{\mathcal{T}_{1}+\mathcal{T}_{2}}}A1subscript𝐴1\scriptstyle{A_{1}}A2subscript𝐴2\scriptstyle{A_{2}}

A model of the coproduct 𝒯 1 + 𝒯 2 subscript 𝒯 1 subscript 𝒯 2 \mathcal{T}_{1}+\mathcal{T}_{2} in a Cartesian Category 𝒞 𝒞 \mathcal{C} is an object X : 𝒞 : 𝑋 𝒞 X:\mathcal{C} that is a model of both 𝒯 1 subscript 𝒯 1 \mathcal{T}_{1} and 𝒯 2 subscript 𝒯 2 \mathcal{T}_{2} .