the empty tuple $\emptyset \to N$ is assigned a morphism $I\to \FFF(N)$ from the monoidal unit.

These data are required to satisfy the following conditions:

1.  for every ${\underline{f}}=(f_1,\dots,f_n) : {\underline{M}}\to N$ and ${\underline{g}}_i =
    (g_{i1},\dots,g_{i k_i}) : {\underline{L}}_i \to M_i$, for $i=1,\dots,n$, the diagram

    \[eqn:FAcomp\] @C=5em[ \_-[((\_1,…,\_n))       ]{} \_[i=1]{}\^n \_[j = 1]{}\^[k\_i]{} (L\_[ij]{}) \^-[\_[i]{} (\_i)]{} & \_[i=1]{}\^n (M\_i)\^-[()]{}\
    &(N) ]{}

    in $\CC$ commutes, where ${\underline{f}}({\underline{g}}_1,\dots,{\underline{g}}_n) :=
    (f_1\,g_{11},\dots,f_n\,g_{nk_n}) : ({\underline{L}}_1,\dots,{\underline{L}}_n)\to N$ is given by composition in $\Loc$;

2.  for every $M\in \Loc$, $\FFF(\id_M) = \id_{\FFF(M)} : \FFF(M)\to \FFF(M) $;

3.  for every ${\underline{f}}=(f_1,\dots,f_n) : {\underline{M}}\to N$ and every permutation $\sigma\in \Sigma_n$, the diagram

    \[eqn:FAperm\] @C=5em[ \_-\_[i=1]{}\^n (M\_i) \^-[()]{} & (N)\
    \_[i=1]{}\^n (M\_[(i)]{}) \_-[  ()]{}& ]{}

    in $\CC$ commutes, where ${\underline{f}}\sigma := (f_{\sigma(1)},\dots,f_{\sigma(n)}) : {\underline{M}}\sigma \to N$ is given by right permutation.

A morphism $\zeta: \FFF\to \GGG$ of prefactorization algebras is a family $\zeta_M : \FFF(M)\to \GGG(M)$ of $\CC$-morphisms, for all $M\in\Loc$, that is compatible with the factorization products, i.e. for all ${\underline{f}} : {\underline{M}}\to N$ the diagram

\[eqn:PFACmorphism\] @C=5em[ \_-[\_i \_[M\_i]{}]{} \_[i=1]{}\^n (M\_i) \^-[ ()]{} & (N) \^-[\_N]{}\
\_[i=1]{}\^n (M\_i) \_-[ ()]{} & (N) ]{}

in $\CC$ commutes.

We denote by $\PFA$ the category of prefactorization algebras on $\Loc$.

Factorization 