How to sum values in excel only if they match a category id in a lookup

Say I have a product category lookup like so:

Sheet 1

Product Name    |    Product Category
--------------------------------------
product 1       |    A
product 2       |    A
product 3       |    B
product 4       |    A
product 5       |    B
product 6       |    C

and I also have a list of purchases which only use Product Name like this:

Sheet 2

Product Name    |    Purchase Quantity
---------------------------------------
product 1       |    35
product 4       |    10
product 5       |    5

I would like to produce a rollup like this:

Product Category   |    Purchase Quantity
------------------------------------------
A                  |    45
B                  |    5
C                  |    0

I've tried a variety of ways to solve this like:

SUMIF(LOOKUP('Sheet 2'!A2:A6,'Sheet 1'!A:A,'Sheet 2'!B:B), "=A", 'Sheet 2'!B2:B6)

SUMPRODUCT(LOOKUP('Sheet 2'!A2:A6, 'Sheet 1'!A:A, 'Sheet 2'!B:B)="A"*'Sheet 2'!B2:B6)

Excel doesn't like the first one. It says the formula is incorrect, but I'm not seeing why. The second one yields #VALUE. Any help on this would be much appreciated. Thanks in advance!

Solution

With A in D2, use this as an array formula.

=SUM(SUMIFS(Sheet2!B$2:B$4, Sheet2!A$2:A$4, IF(Sheet1!B$2:B$7=D2, Sheet1!A$2:A$7)))

Array formulas need to be finished with Ctrl+Shift+Enter, not just Enter.