Code for WooCommerce to check if products in the cart belong to one of the categories we're looking for.

functions.php

<?php
/**
 * Check if a specific product category is in the cart
 */
function wc_ninja_category_is_in_the_cart() {
	// Add your special category slugs here
	$categories = array( 'clothing', 'posters' );

	// Products currently in the cart
	$cart_ids = array();

	// Categories currently in the cart
	$cart_categories = array();

	// Find each product in the cart and add it to the $cart_ids array
	foreach( WC()->cart->get_cart() as $cart_item_key => $values ) {
		$cart_product = $values['data'];
		$cart_ids[]   = $cart_product->id;
	}

	// Connect the products in the cart w/ their categories
	foreach( $cart_ids as $id ) {
		$products_categories = get_the_terms( $id, 'product_cat' );

		// Loop through each product category and add it to our $cart_categories array
		foreach ( $products_categories as $products_category ) {
			$cart_categories[] = $products_category->slug;
		}
	}

	// If one of the special categories are in the cart, return true.
	if ( ! empty( array_intersect( $categories, $cart_categories ) ) ) {
		return true;
	} else {
		return false;
	}
}