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At the beginning of the school year, students are required to take measures that will include the standard University Center’s assessment battery.

Should it be Center's or Center?

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Either can work. Both forms convey essentially the same information.

With the possessive, the phrase is understood in the straightforward manner: it's the assessment battery related to (owned by / administered by / etc) the University Center.

Without the possessive, it makes University Center an attributive noun.

In grammar, a noun adjunct or attributive noun or noun (pre)modifier is an optional noun that modifies another noun; it is a noun functioning as a pre-modifier in a noun phrase. For example, in the phrase "chicken soup" the noun adjunct "chicken" modifies the noun "soup". - wikipedia

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