Here come a few more questions, I apologize if they were already asked during the course. Again, trying to make things more or less precise, but may fail in that :)

1. Let's say we have a category \\(C\\), which has all products. Is it true that we can define a functor \\(\otimes: C \times C \to C\\) by pointing each pair \\((x, y)\\) to the product of \\(x\\) and \\(y\\), \\(x \leftarrow p \rightarrow y\\), and in this way make \\(C\\) somewhat monoidal?

2. Working with the same category \\(C\\), which happens to have all coproducts, we define a functor \\(\oplus: C \times C \to C\\) by sending each pair \\((x, y)\\) to their coproduct \\(x \rightarrow p \leftarrow y\\). How this structure (\\(C, \oplus)\\) is called then?

3. For any other structure \\((C, \odot, \boxdot)\\) with \\(\odot: C \times C \to C\\) and \\(\boxdot: C \times C \to C\\), where we are using spans and cospans of pairs of elements \\((x, y)\\) to define \\(\odot, \boxdot\\), is it true that we can always establish a pair of adjoints between \\((C, \otimes, \oplus)\\) and \\((C, \odot, \boxdot)\\), using the property of products and coproducts being limits and colimits (so we always can move from a span to a limit, and from a colimit to a cospan in a natural way). So these structures \\((C, \odot, \boxdot)\\) may be viewed as more relaxed versions of \\((C, \otimes, \oplus)\\)?

4. Are there any other functor \\(OP: C \times C \to C\\), which is interesting in a certain way, so we can define a structure \\((C, \otimes, \oplus, OP)\\)? The way \\(\otimes, \oplus\\) are defined in 1 and 2 makes them interesting, are there any other universal objects, except for products and coproducts, which can be utilized to construct \\(OP\\) as well?

1. Let's say we have a category \\(C\\), which has all products. Is it true that we can define a functor \\(\otimes: C \times C \to C\\) by pointing each pair \\((x, y)\\) to the product of \\(x\\) and \\(y\\), \\(x \leftarrow p \rightarrow y\\), and in this way make \\(C\\) somewhat monoidal?

2. Working with the same category \\(C\\), which happens to have all coproducts, we define a functor \\(\oplus: C \times C \to C\\) by sending each pair \\((x, y)\\) to their coproduct \\(x \rightarrow p \leftarrow y\\). How this structure (\\(C, \oplus)\\) is called then?

3. For any other structure \\((C, \odot, \boxdot)\\) with \\(\odot: C \times C \to C\\) and \\(\boxdot: C \times C \to C\\), where we are using spans and cospans of pairs of elements \\((x, y)\\) to define \\(\odot, \boxdot\\), is it true that we can always establish a pair of adjoints between \\((C, \otimes, \oplus)\\) and \\((C, \odot, \boxdot)\\), using the property of products and coproducts being limits and colimits (so we always can move from a span to a limit, and from a colimit to a cospan in a natural way). So these structures \\((C, \odot, \boxdot)\\) may be viewed as more relaxed versions of \\((C, \otimes, \oplus)\\)?

4. Are there any other functor \\(OP: C \times C \to C\\), which is interesting in a certain way, so we can define a structure \\((C, \otimes, \oplus, OP)\\)? The way \\(\otimes, \oplus\\) are defined in 1 and 2 makes them interesting, are there any other universal objects, except for products and coproducts, which can be utilized to construct \\(OP\\) as well?