🍕(--, B) : C -> Set denotes the contravariant hom functor, normally written Hom(--, B). In this case, C is a category, and B is a fixed object in that category. The -- can be replaced by either an object or morphism of C, and that defines a map from C to Set.

For any given object X in C, the hom-set Hom(X, C) is the set of morphisms X -> B in C. For a morphism f : X -> Y in C, the Set morphism Hom(f, B) : Hom(Y, B) -> Hom(X, B) is defined by sending each g : Y -> B to gf : X -> B. This is the mapping C -> Set defined by Hom(--, C), and it's a (contravariant) functor because it respects composition: if h : X -> Y and f : Y -> Z then fh : X -> Z and Hom(fh, C) = Hom(h, C)Hom(f, C) sends g : Z -> B to gfh : X -> B.

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P^(n)(R) AKA RP^n is the n-dimensional real projective space.

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The caveat "phi is a morphism" is probably just to clarify that we're talking about "all morphisms X -> Y [in a given category]" and not simply all functions or something.

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For more context, the derived functor of Hom(--, B) is called the Ext functor, and the exactness of that sequence (if the typo were fixed) is the statement of the universal coefficient theorem (for cohomology): https://en.wikipedia.org/wiki/Universal_coefficient_theorem The solution to this problem is the "Example: mod 2 cohomology of the real projective space" on that page. It's (Z/2Z)[x] / <x^(n+1)> or 🍔[x]/<x^(n+1)>, i.e. the ring of polynomials of degree n or less with coefficients in 🍔 = Z/2Z, meaning coefficients of 0 or 1.