Let ( E α , f β α ) (\mathrm{E}_\alpha, f_{\beta\alpha}) ( E α , f β α ) ( F α , g β α ) (\mathrm{F}_\alpha, g_{\beta\alpha}) ( F α , g β α ) ( G α , h β α ) (\mathrm{G}_\alpha, h_{\beta\alpha}) ( G α , h β α ) be three direct systems of sets relative to I. Let E = l i m → E α \mathrm{E} = \varinjlim \mathrm{E}_\alpha E = lim E α F = l i m → F α \mathrm{F} = \varinjlim \mathrm{F}_\alpha F = lim F α G = l i m → G α \mathrm{G} = \varinjlim \mathrm{G}_\alpha G = lim G α and let f α f_\alpha f α (resp. g α g_\alpha g α h α h_\alpha h α E α \mathrm{E}_\alpha E α (resp. F α \mathrm{F}_\alpha F α G α \mathrm{G}_\alpha G α (resp. F, G*). If* ( u α ) (u_\alpha) ( u α ) and ( v α ) (v_\alpha) ( v α ) are two direct systems of mappings u α : E α → F α u_\alpha : \mathrm{E}_\alpha \to \mathrm{F}_\alpha u α : E α → F α v α : F α → G α v_\alpha : \mathrm{F}_\alpha \to \mathrm{G}_\alpha v α : F α → G α then the mappings v α ∘ u α : E α → G α v_\alpha \circ u_\alpha : \mathrm{E}_\alpha \to \mathrm{G}_\alpha v α ∘ u α : E α → G α form a direct system of mappings, and we have
(25)
l i m → ( v α ∘ u α ) = ( l i m → v α ) ∘ ( l i m → u α ) . \varinjlim (v_\alpha \circ u_\alpha) = (\varinjlim v_\alpha) \circ (\varinjlim u_\alpha). lim ( v α ∘ u α ) = ( lim v α ) ∘ ( lim u α ) . For if we put w α = v α ∘ u α w_\alpha = v_\alpha \circ u_\alpha w α = v α ∘ u α α ⩽ β \alpha \leqslant \beta α ⩽ β
h β α ∘ w α = ( h β α ∘ v α ) ∘ u α = ( v β ∘ g β α ) ∘ u α = v β ∘ ( u β ∘ f β α ) = w β ∘ f β α , h_{\beta\alpha} \circ w_\alpha = (h_{\beta\alpha} \circ v_\alpha) \circ u_\alpha = (v_\beta \circ g_{\beta\alpha}) \circ u_\alpha = v_\beta \circ (u_\beta \circ f_{\beta\alpha}) = w_\beta \circ f_{\beta\alpha}, h β α ∘ w α = ( h β α ∘ v α ) ∘ u α = ( v β ∘ g β α ) ∘ u α = v β ∘ ( u β ∘ f β α ) = w β ∘ f β α , which shows that ( w α ) (w_\alpha) ( w α ) u = l i m → u α u = \varinjlim u_\alpha u = lim u α v = l i m → v α v = \varinjlim v_\alpha v = lim v α α ∈ I \alpha \in \mathrm{I} α ∈ I
( v ∘ u ) ∘ f α = v ∘ ( g α ∘ u α ) = h α ∘ ( v α ∘ u α ) , (v \circ u) \circ f_\alpha = v \circ (g_\alpha \circ u_\alpha) = h_\alpha \circ (v_\alpha \circ u_\alpha), ( v ∘ u ) ∘ f α = v ∘ ( g α ∘ u α ) = h α ∘ ( v α ∘ u α ) , and by virtue of the uniqueness of the direct limit, we have v ∘ u = l i m → w α v \circ u = \varinjlim w_\alpha v ∘ u = lim w α
Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.
Nothing.