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Several people have mentioned that $\otimes N$ should left adjoint to a suitable $Hom(N,-)$. There are numerous instances of this sort of construction (including one using joins of augmented simplicial sets mentioned in another question w.r.t Lurie's HTT). There is a good discussion of the ideas in MacLane's Homology and in Hilton and Stammbach. The sign conventions are, however, often not fully explained. (They work, but why? is the feeling one gets.)

The point in Baez and Crans is more that there are variants that have good more or less geometric interpretations and they give sensible answers. (There is also work by Andy Tonks on tensor products of crossed complexes that makes this clear.)

The discussion in older sources such as Spanier is also good for explaining why the signs are important.

You ask about bilinearity. This is really a strange idea. It can be handled in more generality than the simple form you mention but it can often be best interpreted as coming from the evaluation map $Hom(A,B)\times A \to B$ in which $(f,x)$ gets sent to $f(x)$. If you look at the adjunction that was mentioned between $\otimes$ and $Hom$, then from $\phi : C\to Hom(A,B)$ you do a naive thing and get $C\times A \to Hom(A,B)\times A\to B$ and that will be bilinear, ugh! Thus the representability of bilinear maps is the same universal property as the one everyone has been mentioning.

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Several people have mentioned that $\otimes N$ should left adjoint to a suitable $Hom(N,-)$. There are numerous instances of this sort of construction (including one using joins of augmented simplicial sets mentioned in another question w.r.t Lurie's HTT). There is a good discussion of the ideas in MacLane's Homology and in Hilton and Stammbach. The sign conventions are, however, often not fully explained. (They work, but why? is the feeling one gets.)

The point in Baez and Crans is more that there are variants that have good more or less geometric interpretations and they give sensible answers. (There is also work by Andy Tonks on tensor products of crossed complexes that makes this clear.)

The discussion in older sources such as Spanier is also good for explaining why the signs are important.