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The first "claim"(seeming) claim is false for the functor $F(M)=M\otimes M$. If $M$ is connected then $F(M)$ is $1$-connected, but of course $F_2(M,M)=F(M)\oplus F(M)$ will never be any more highly connected than $F(M)$.
The first "claim" is false for the functor $F(M)=M\otimes M$. If $M$ is connected then $F(M)$ is $1$-connected, but of course $F_2(M,M)=F(M)\oplus F(M)$ will never be any more highly connected than $F(M)$.
The first (seeming) claim is false for the functor $F(M)=M\otimes M$. If $M$ is connected then $F(M)$ is $1$-connected, but of course $F_2(M,M)=F(M)\oplus F(M)$ will never be any more highly connected than $F(M)$.
The first "claim" is false for the functor $F(M)=M\otimes M$. If $M$ is connected then $F(M)$ is $1$-connected, but of course $F_2(M,M)=F(M)\oplus F(M)$ will never be any more highly connected than $F(M)$.
