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$$\begin{array}{ccccccccc}& & (\mathrm{\infty },1)\mathrm{Sh}(C)& & & (\mathrm{\infty },1)\mathrm{PSh}(C)& & & (\mathrm{\infty },1)\mathrm{Sh}(C)\\ & & {\uparrow }^{\mathrm{presentation}}& & & {\uparrow }^{\mathrm{presentation}}& & & {\uparrow }^{\mathrm{presentation}}\\ \mathrm{SSh}(C{)}_{\mathrm{inj}}^{l\mathrm{loc}}& \stackrel{\stackrel{\mathrm{sheafification}}{\leftarrow }}{\stackrel{\mathrm{embedding}}{\to }}& \mathrm{SPSh}(C{)}_{\mathrm{inj}}^{l\mathrm{loc}}& \stackrel{}{\leftarrow }\mid & \mathrm{SPSh}(C{)}_{\mathrm{inj}}& \stackrel{\stackrel{\mathrm{Id}}{\leftarrow }}{\stackrel{\mathrm{Id}}{\to }}& \mathrm{SPSh}(C{)}_{\mathrm{proj}}& \stackrel{}{\mapsto }& \mathrm{SPSh}(C{)}_{\mathrm{proj}}^{l\mathrm{loc}}& \stackrel{\stackrel{\mathrm{sheafification}}{\to }}{\stackrel{\mathrm{embedding}}{\leftarrow }}& \mathrm{SSh}(C{)}_{\mathrm{proj}}^{l\mathrm{loc}}\\ \mathrm{Joyal}& \stackrel{\mathrm{Quillen}\mathrm{equivalence}}{\leftrightarrow }& \mathrm{Jardine}& \stackrel{\mathrm{left}\mathrm{Bousf}.\mathrm{localization}}{\leftarrow \mid }& \mathrm{Heller}& \stackrel{\mathrm{Quillen}\mathrm{equivalence}}{\leftrightarrow }& \mathrm{Bousfield}-\mathrm{Kan}& \stackrel{\mathrm{left}\mathrm{Bousf}.\mathrm{localization}}{\mapsto }& \mathrm{Blander}& \stackrel{\mathrm{Quillen}\mathrm{equivalence}}{\leftrightarrow }& \mathrm{Brown}-\mathrm{Gersten}\\ \\ & \mathrm{everything}\mathrm{cofibrant};\\ & \mathrm{fibrant}=\mathrm{global}\mathrm{injective}\mathrm{fib}...\\ & ...\mathrm{satisfying}\mathrm{descent}& & & & & & & & \mathrm{cofibrant}=\mathrm{global}\mathrm{projective}\mathrm{cofib};\\ & & & & & & & & & \mathrm{fibrant}=\mathrm{Kan}\mathrm{valued}\mathrm{and}...\\ & & & & & & & & & ...\mathrm{satisfying}\mathrm{descent}\end{array}$$\array{ && (\infty,1)Sh(C) &&& (\infty,1)PSh(C) &&& (\infty,1)Sh(C) \\ && \uparrow^{presentation} &&& \uparrow^{presentation} &&& \uparrow^{presentation} \\ SSh(C)^{l loc}_{inj} & \stackrel{\stackrel{sheafification}{\leftarrow}} {\stackrel{embedding}{\to}}& SPSh(C)^{l loc}_{inj} &\stackrel{}{\leftarrow}|& SPSh(C)_{inj} &\stackrel{\stackrel{Id}{\leftarrow}} {\stackrel{Id}{\rightarrow}}& SPSh(C)_{proj} &\stackrel{}{\mapsto}& SPSh(C)_{proj}^{l loc} & \stackrel{\stackrel{sheafification}{\to}} {\stackrel{embedding}{\leftarrow}}& SSh(C)_{proj}^{l loc} \\ Joyal &\stackrel{Quillen equivalence}{\leftrightarrow}& Jardine &\stackrel{left Bousf. localization}{\leftarrow|}& Heller &\stackrel{Quillen equivalence}{\leftrightarrow}& Bousfield-Kan &\stackrel{left Bousf. localization}{\mapsto}& Blander &\stackrel{Quillen equivalence}{\leftrightarrow}& Brown-Gersten \\ \\ & everything cofibrant; \\ & fibrant = global injective fib... \\ \;\;\; & ...satisfying descent &&&&&&&& cofibrant = global projective cofib; \\ &&&&&&&&& fibrant = Kan valued and... \\ &&&&&&&&& \;\;\; ...satisfying descent }