Operations on pullbacks of vector bundles. I wanted to prove the following bundle isomorphisms, on page 20 of hatcher's $K$-theroy. That is 
$$f^*(E_1) \otimes f^*(E_2) \cong f^*(E_1 \otimes E_2)$$ 
$$f^*(E_1) \oplus f^*(E_2) \cong f^*(E_1 \oplus E_2)$$ 
So I want to show they satisfy the UP. Indeed, there is a well defined map, for each $i =1,2$, 
$$ f^*(E_i) \xrightarrow{f'_i} E_i$$
Then fiberwise we have a linear isomorphism of (say for example) 
$$ f^*(E_1) \otimes f^*(E_1) \xrightarrow {f'_1 \otimes f'_2} E_1 \otimes E_2$$ 
This map solves the universal property of pull back, except that it is defined fiberwise so we haven't proved it is continuous. How do I prove this? 

Thoughts: I think this follows from how we define the topology. From my previous question, we interpret, with $B$ as base space,  $$E_1 \otimes E_2 := \bigsqcup_{x \in B} (E_1)_x \otimes (E_2)_x. $$ 
and a basis open set is of the form 
$$ \bigsqcup_{x \in U} (E_1)_x \otimes (E_2)_x$$ 
where $U$ is taken to be a common intersection of the coverings from spaces $E_i$. 
So the preimage of this set under $f'_1 \otimes f'_2$ is 
$$ \bigsqcup_{y \in f^{-1}(U)} (f^*(E_1)_y \otimes f^*(E_2)_y)$$ 
which is an open set. Hence, as we have shown this for basis elements, the full map is continuous. 
 A: Your thoughts concerning topologizing the bundles are correct. After the proof of Proposition 1.5 Hatcher gets explicit about local trivializations of the pullback bundle $f^*(E)$, and that is all you need. For each $x \in B$ choose an open neighborhood $U$ such that both $E_1, E_2$ (and hence also $E_1 \otimes E_2$) are trivial over $U$. This canonically induces trivializations of $f^*(E_1),f^*(E_2), f^*(E_1 \otimes E_2)$ over $f^{-1}(U)$. With respect to these trivializations the fiberwise isomorphism $f^*(E_1) \otimes f^*(E_1) \to E_1 \otimes E_2$ corresponds on $f^{-1}(U)$ to $f \times id : f^{-1}(U) \times (\mathbb{R}^{n_1} \otimes \mathbb{R}^{n_2}) \to  U \times (\mathbb{R}^{n_1} \otimes \mathbb{R}^{n_2})$ which is obviously is continuous. Hence $f^*(E_1) \otimes f^*(E_1) \to E_1 \otimes E_2$ is locally continuous which suffices to prove the desired result.
Another way to see it is to consider the obvious fiberwise isomorphism $f^*(E_1) \otimes f^*(E_1) \to f^*(E_1 \otimes E_2)$ of bundles over $A$. On $f^{-1}(U)$ it corresponds to the identity on $f^{-1}(U) \times (\mathbb{R}^{n_1} \otimes \mathbb{R}^{n_2})$. Hence it is locally a bundle isomorphism which again suffices to prove the result.
By the way, perhaps you should consult also other books. Hatcher is a little short when dealing with the basic material. For example, he does not mention the relationship of bundles with the transition maps $U_\alpha \cap U_\beta \to GL_n(\mathbb{R})$ associated to a bundle atlas (see Andres Mejia's comment).
Here are some books:
Steenrod, Norman Earl. The topology of fibre bundles. Vol. 14. Princeton University Press, 1999
Husemoller, Dale. Fibre bundles. Vol. 5. New York: McGraw-Hill, 1966. https://www.maths.ed.ac.uk/~v1ranick/papers/husemoller
Cohen, Ralph. The Topology of Fiber Bundles. http://math.stanford.edu/~ralph/fiber.pdf
Atiyah, Michael. K-theory. CRC Press, 2018. https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf
