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?